Measuring temperature & physical symptoms is a good start. It would be great if these trials would measure each recipients HLA SNPs (see GSK MERS 2009 fiasco and others), T-cells (CD4/8/57/etc), inflammation markers like C3a,C4a,TGF-b, MMP-9, and cytokines like TNF-a -- before and after each vaccine dose. Measuring for all ingredients pre-post each dose is also important because the metabolization for each component is not uniform. Stop using 2010+ technology to make vaccines & medications and then evaluating the applied result with 1990s (or earlier) methods.
So if you want to shut up 'anti-vaxxers' than collect/share the data from everyone who receives a vaccine and improve the process instead of shrouding results or conducting half-assed studies (AstraZeneca using meningitis vaccine as control instead of true placebo).